We prove anti-concentration results for polynomials of independent
random variables with arbitrary degree. Our results extend the
classical Littlewood-Offord result for linear polynomials, and improve
several earlier estimates.

We discuss applications in two different areas. In complexity theory,
we prove near-optimal lower bounds for computing the PARITY function,
addressing a challenge in complexity theory posed by Razborov and
Viola, and also address a problem concerning the OR function. In
random graph theory, we derive a general anti-concentration result on
the number of copies of a fixed graph in a random graph.